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2003 | Buch

Adaptive Finite Element Methods for Differential Equations

verfasst von: Wolfgang Bangerth, Rolf Rannacher

Verlag: Birkhäuser Basel

Buchreihe : Lectures in Mathematics ETH Zürich

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Über dieses Buch

These Lecture Notes have been compiled from the material presented by the second author in a lecture series ('Nachdiplomvorlesung') at the Department of Mathematics of the ETH Zurich during the summer term 2002. Concepts of 'self­ adaptivity' in the numerical solution of differential equations are discussed with emphasis on Galerkin finite element methods. The key issues are a posteriori er­ ror estimation and automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method (or shortly D WR method) for goal-oriented error estimation is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. 'Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. The basics of the DWR method and various of its applications are described in the following survey articles: R. Rannacher [114], Error control in finite element computations. In: Proc. of Summer School Error Control and Adaptivity in Scientific Computing (H. Bulgak and C. Zenger, eds), pp. 247-278. Kluwer Academic Publishers, 1998. M. Braack and R. Rannacher [42], Adaptive finite element methods for low­ Mach-number flows with chemical reactions.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
We begin with a brief introduction to the philosophy underlying the approach to self-adaptivity which will be discussed in these Lecture Notes.
Wolfgang Bangerth, Rolf Rannacher
Chapter 2. An ODE Model Case
Abstract
In the following, we consider the realization of the ideas sketched in the Introduction for the initial value problem of an autonomous ODE system:
$$u'(t) = f(u(t)),t \in I: = [0,T],u(0) = {u_0}$$
(2.1)
.
Wolfgang Bangerth, Rolf Rannacher
Chapter 3. A PDE Model Case
Abstract
In this chapter, we will develop the basics of the DWR method for linear elliptic partial differential equations as originally described in Becker and Rannacher [30].
Wolfgang Bangerth, Rolf Rannacher
Chapter 4. Practical Aspects
Abstract
In this chapter, we discuss several aspects of the practical use of the DWR method described in the previous sections. These are (i) the practical and efficient evaluation of the a posteriori error representations, (ii) the extraction of local refinement indicators, and (iii) the design of strategies for economical mesh adaptation.
Wolfgang Bangerth, Rolf Rannacher
Chapter 5. The Limits of Theoretical Analysis
Abstract
In this chapter, we want to discuss some questions concerning the theoretical justification of the DWR method for goal-oriented mesh adaptivity presented so far. We will see that this task is rather demanding and poses several new questions for the theoretical analysis of the finite element method. In fact, relying on the available results from the literature, we do not reach very far, yet. Since several not very practical assumptions will be used, we dispense with stating formal propositions.
Wolfgang Bangerth, Rolf Rannacher
Chapter 6. An Abstract Approach for Nonlinear Problems
Abstract
In this chapter, we will present a very general approach to a posteriori error estimation for the Galerkin approximation of nonlinear variational problems as developed in Becker and Rannacher [31]. The framework is kept on an abstract level in order to allow later for a unified application to rather different situations, such as nonlinear PDEs, but also eigenvalue and optimization problems.
Wolfgang Bangerth, Rolf Rannacher
Chapter 7. Eigenvalue Problems
Abstract
In the following, we will apply the abstract theory of the DWR method developed in Chapter 6 to error control in the approximation of eigenvalue problems.
Wolfgang Bangerth, Rolf Rannacher
Chapter 8. Optimization Problems
Abstract
As another important application for the general theory of the DWR method developed in Chapter 6, we consider optimization problems with PDE constraints as discussed in Kapp [96] and Becker et al. [28]. In abstract variational notation, such problems are posed in a state space V and a control space Q , on which state and control operators, associated with the forms A(·)(·) and B(·, ·), respectively, act and on which the cost functional J(·.·) is defined.
Wolfgang Bangerth, Rolf Rannacher
Chapter 9. Time-Dependent Problems
Abstract
In this chapter, we will apply the general theory developed in the preceding sections to evolution problems of the following general form: Find uV satisfying
(9.1)
.
Wolfgang Bangerth, Rolf Rannacher
Chapter 10. Applications in Structural Mechanics
Abstract
In this chapter, we will present some applications of the DWR method to typical problems in structural mechanics. At first, the standard finite element approximation of the linear Lamé-Navier system is considered which is the basic model for small static deformations of elastic bodies. This does not add much to the experience we have already gained before at the Poisson problem. Then, we turn to a more challenging problem, the deformation of an elasto-plastic body assuming linear-elastic and perfectly plastic material. The finite element approximation is rather standard but includes stabilization in order to cope with almost incompressible material behavior. Here, the interesting point is the non-differentiable nonlinearity which occurs if the constrained problem is reformulated as a variational equation. The results presented in this Chapter are taken from Suttmeier [128], and Rannacher and Suttmeier [118, 119, 120, 121].
Wolfgang Bangerth, Rolf Rannacher
Chapter 11. Applications in Fluid Mechanics
Abstract
In this chapter, we apply the DWR method to problems in fluid mechanics which are all related to the ‘incompressible’ Navier-Stokes equation for pairs u = {v,p} :
$$A(u): = \left[ \begin{gathered} - v\Delta v + v\cdot\nabla u + \nabla p - f \hfill \\ \nabla \hfill \\ \end{gathered} \right] = 0$$
.
Wolfgang Bangerth, Rolf Rannacher
Chapter 12. Miscellaneous and Open Problems
Abstract
The material presented in these Lecture Notes demonstrates that duality-based error estimation as realized in the DWR method can be applied, in principle, to all problems posed in variational form, even if their regularity properties do not match all assumptions. The additional work for the adaptive component of the solution process is relatively small. In fact, for nonlinear problems the evaluation of the a posteriori error estimates amounts to about the equivalent of one extra step within the outer Newton iteration on the current mesh. In this case, the extra work for mesh adaptation usually makes up 5–25% of the total work on the optimized mesh. However, the implementation may appear difficult when existing software components like mesh generators, multigrid solvers, etc., cannot be used directly.
Wolfgang Bangerth, Rolf Rannacher
Backmatter
Metadaten
Titel
Adaptive Finite Element Methods for Differential Equations
verfasst von
Wolfgang Bangerth
Rolf Rannacher
Copyright-Jahr
2003
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-7605-6
Print ISBN
978-3-7643-7009-1
DOI
https://doi.org/10.1007/978-3-0348-7605-6